1.4 Estimating Limit Values from Tables

• The limit of a function is the value the function reaches at a specific input value.

  False
• A function maps an input to a single output
• The limit of a function describes the value the function approaches as its input gets closer to a certain point
• What is the limit of \( f(x) \) as \( x \) approaches 3 based on the table in line 39?

  6
• In the table where \( x = 2 \), the function value \( f(2) \) is 5.


  True
• If \( x = 1 \), the function value \( f(x) \) is 3 according to the table in line 223.

  True
• What are two possible trends to observe in function values as they approach a limit?

  Increasing or decreasing
• What is the first step in estimating the limit of a function as x approaches a specific value?

  Identify trends
• What does the limit of a function describe?

  Approaching a specific value
• What is the first step in interpreting function values from a table?

  Identify the input value
• What are two possible directions that function values can approach a limit from?

  Above or below
• Steps to identify patterns and estimate limit values using a table

  1️⃣ Look for trends in how function values change

  2️⃣ Observe if values are increasing or decreasing

  3️⃣ Check if approaching from above or below

  4️⃣ Estimate the limit value
• Steps to estimate the limit of a function using a table

  1️⃣ Examine the table for \( x \) values near \( a \)

  2️⃣ Identify trends in function values

  3️⃣ Determine the value \( L \) that \( f(x) \) approaches
• The left-hand limit is denoted as $\lim_{x \to a^ - } f(x) =$$L$ and represents the value \( f(x) \) approaches as \( x \) approaches \( a \) from values less than \( a
• If the left-hand and right-hand limits both equal \( L \), the overall limit exists and equals L
• Steps to determine if a limit exists

  1️⃣ Check if the left-hand limit exists

  2️⃣ Check if the right-hand limit exists

  3️⃣ Determine if left-hand and right-hand limits are equal

  4️⃣ If equal, the limit exists; otherwise, it does not
• How is the limit of a function written mathematically as x approaches a?

  $\lim_{x \to a} f(x) =$$L$
• Steps to interpret function values from a table

  1️⃣ Identify the input value \( x \)

  2️⃣ Locate the corresponding output value \( f(x) \)

  3️⃣ State the function value at that input
• What is the limit of \( f(x) \) as \( x \) approaches 3 based on the table in line 178?

  6
• Identifying patterns in a table helps estimate the limit
• Function values can approach a limit from above or below
• When estimating limits, observe if the function values are increasing or decreasing
• The notation for the limit of a function \( f(x) \) as \( x \) approaches \( a \) is L
• To find \( f(2) \) in a table, locate the row where \( x = 2 \) and find the corresponding output
• If function values are approaching the limit from below, the limit value is likely just above
• If function values increase and approach the limit from below, the limit value is likely just above the last function value in the table
• As \( x \) approaches 2, \( f(x) \) approaches 4
• In the example where \( x \) approaches 2, \( f(x) \) approaches 4
• Tables can represent a function by showing pairs of input values and their corresponding output
• When estimating a limit, it is important to observe if the function values are increasing or decreasing.


  True
• If function values are increasing and approaching the limit from below, the limit value is likely just above the last function value in the table.


  True
• The second step in estimating a limit is to observe whether the function values approach the limit from above or below.


  True
• A function maps each input value to a single output value.

  True
• To estimate limits using tables, it is necessary to look for trends in how function values change as input values get closer to the point of interest.

  True
• As \( x \) approaches 3, \( f(x) \) approaches 6
• As \( x \) approaches 3, \( f(x) \) approaches 6.


  True
• As \( x \) approaches 2, \( f(x) \) approaches 4.


  True
• Left-hand and right-hand limits help understand function behavior near a point.


  True
• The overall limit exists if left-hand and right-hand limits are equal.


  True
• If left-hand and right-hand limits differ, the overall limit does not exist.


  True